Download COMPLEMENTARY AND SUPPLEMENTARY ANGLES LESSON 2

COMPLEMENTARY AND SUPPLEMENTARY
ANGLES
LESSON 2-D
I
ndividual angles are classified as acute, right, obtuse or straight. Special pairs of angles can also be classified.
In this lesson two special pairs of angles and their relationships will be examined.
EXPLORE!
COMPLEMENTARY VS SUPPLEMENTARY
Step 1: Look at the angles in the chart.
a. What similarities do you notice about the pairs of angles called supplementary angles?
b. What similarities do you notice about the pairs of angles called complementary angles?
SUPPLEMENTARY ANGLES
2
∠1 and ∠2 are
supplementary.
1
COMPLEMENTARY ANGLES
R
M
E
O
C 40° T
A
D
140°
O
G
L
E
R
F
T
G
m∠6 = 80° and m∠7 = 100°
∠CAT and ∠DOG
are supplementary.
∠6 and ∠7 are
supplementary.
Step 2: Write a definition for supplementary angles.
Step 3: Write a definition for complementary angles.
Step 4: Give at least two examples of angle measures for
each type of angle pair listed below.
a. Supplementary angles
b. Complementary angles
14 R
C
70°
A
∠LEF and ∠RGT are
supplementary.
Lesson 2-D ~ Complementary And Supplementary Angles
2
∠ROM and ∠MOE
are complementary.
W
20°
K
1
m∠8 = 45° and m∠9 = 45°
L
∠CAR and ∠WLK
are complementary.
∠1 and ∠2 are
complementary.
∠8 and ∠9 are
complementary.
Complementary and supplementary angles are special pairs of angles. Complementary angles are two angles
with a sum of 90°. Two angles with a sum of 180° are called supplementary angles. These special pairs of
angles may or may not be adjacent. Adjacent angles share a side.
EXAMPLE 1
Use the diagram to find m∠PAR.
R
45°
Solution
m∠PAR and m∠TAR form a straight
angle; therefore they are supplementary.
P
A
T
Supplementary angles have a sum of 180°.
m∠PAR + 45° = 180°
Subtract 45 from both sides of the equation.
−45° −45° m∠PAR = 135°
☑ Check the solution.
m∠PAR is 135°
EXAMPLE 2
135° + 45° =? 180°
180° = 180°
∠GRA and ∠INS are supplementary.
G
°
a. Write an equation to solve for x.
4) A
+
I
x
b. Determine the measure of each angle.
(2
R
(3x + 1)°
N
Solutions
S
a. Supplementary angles have a sum of 180°.
m∠GRA + m∠INS = 180°
Write an equation to solve for x.
(2x + 4) + (3x + 1) = 180
Combine like terms.
5x + 5 = 180
Subtract 5 from each side.
−5 −5 5x
__
Divide each side by 5.
​    ​ = ___
​ 175
   ​ 5
5
x = 35
b. Write the given
expression for each angle. m∠GRA = (2x + 4)° m∠INS = (3x + 1)°
Substitute 35 for x. = (2(35) + 4)°
= (3(35) + 1)°
Multiply.
= (70 + 4)°
= (105 + 1)°
Add.
= 74°
= 106°
☑ m∠GRA + m∠INS = 180°
?
74° + 106° = 180°
180° = 180°
The measure of ∠GRA is 74°.
The measure of ∠INS is 106°.
Lesson 2-D ~ Complementary And Supplementary Angles 15
EXAMPLE 3
Use the diagram to write an equation. Solve for x.
E
O
62°
(x + 5)°
M
H
Solution
Complementary angles have a sum of 90°.
m∠HOM + m∠OEM = 90°
Substitute the degreee measures.
62 + (x + 5) = 90
Combine like terms.
x + 67 = 90
Subtract 67 from each side of the equation. −67 −67
x = 23
☑ Check the solution.
62 + (23 + 5) =? 90
62 + 28 =? 90
90 = 90
The value of x is 23.
EXAMPLE 4
∠1 and ∠2 are complementary angles. The measure of ∠1 = (3x + 4)°
and m∠2 = (x + 6)°.
a. Draw a diagram.
b. Write an equation and solve for x.
c. Find m∠1 and m∠2.
Solutions
a. (x + 6)°
(3x + 4)°
∠1
∠2
or
°
4)
+
x
(3
(x + 6)°
b. Complementary angles have a sum of 90°. m∠1 + m∠2 = 90°
Substitute the degree measures.
(3x + 4) + (x + 6) = 90
Combine like terms.
4x + 10 = 90
Subtract 10 from each side of the equation.
−10 −10
__
Divide by 4 on each side of the equation.
​ 4x
  ​ = __
​  80
  ​ 4
4
x = 20
c. Write the given expression
for each angle.
Substitute 20 for x. Multiply. Add.
m∠1 = (3x + 4)° = (3(20) + 4)
= (60 + 4)
= 64°
☑ Check the solution.
m∠1 = 64° and m∠2 = 26°.
16 Lesson 2-D ~ Complementary And Supplementary Angles
m∠2 = (x + 6)°
= (20 + 6)
= 26°
?
64° + 26° = 90°
90 = 90
EXERCISES
Identify each pair of angles as complementary, supplementary or neither.
1.
2.
56°
110°
34°
3.
4.
5.
6.
122°
70°
52°
38°
58°
20°
122°
7. m∠1 and m∠2 sum to 181°.
8. ∠A and ∠M have a sum of 90°.
Write an equation for each description. Solve for x. Check your solution.
9. ∠A and ∠B are complementary
42°
B
A
10.
x°
x°
11.
41°
12.
)°
x°
70° (4 + 2x)°
13.
x°
14.
5
x−
7
(
(x + 3)°
2x° x°
Lesson 2-D ~ Complementary And Supplementary Angles 17
Write an equation for each description. Solve for x. Check your solution.
15. ∠MAN and ∠MAP are supplementary. The measure of ∠MAN is 57°. What is the measure of ∠MAP?
16. ∠5 and ∠7 are complementary angles. Find the measure of ∠7 if m∠5 = 47°.
17. The complement of ∠Q is 31°. Find m∠Q.
18. The supplement of ∠U is 62°. What is m∠U?
Find the measure of each angle in Exercises 19-23. Check your solution.
19. ∠1 and ∠2 are supplementary; m∠1 = 3x° and m∠2 = 3x°.
20. ∠W and ∠C are complementary; m∠W = (47 + 3x)° and m∠C = (10 + 8x)°.
21. ∠G and ∠H are supplementary; m∠G = (x + 4)° and m∠H = (4x + 11)°.
22. ∠V and m∠W are supplementary; ∠V = (12 + 3x)° and m∠W = (33 + 2x)°.
23. ∠1 and ∠2 are complementary; m∠1 = 4x° and m∠2 = (x + 8)°.
3 5
11
24. Use the figure on the right.
18 a. Name an angle that appears to be obtuse.
b. Name three pairs of supplementary angles.
c. Name a pair of complementary angles.
d. Give possible measures for ∠4 and ∠1.
e. Determine m∠11.
Lesson 2-D ~ Complementary And Supplementary Angles
10
4
1
2
7 6
8 9